Margins Study Assistant

Chapter 1

Sampling and data

Summary

Statistics begins by naming what you are studying. A population is the whole group, a sample is the part you actually observe, a parameter describes the population, and a statistic describes the sample. Data are qualitative or quantitative, and quantitative data are discrete or continuous. Levels run from nominal labels through ordinal, interval, and ratio. How you sample — simple random, stratified, cluster, systematic, or convenience — decides whether the result can be trusted.

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Average: also called mean; a number that describes the central tendency of the data
Blinding: not telling participants which treatment a subject is receiving
Categorical Variable: variables that take on values that are names or labels
Cluster Sampling: a method for selecting a random sample and dividing the population into groups (clusters); use simple random sampling to select a set of clusters. Every individual in the chosen clusters is included in the sample.
Continuous Random Variable: a random variable (RV) whose outcomes are measured; the height of trees in the forest is a continuous RV.
Control Group: a group in a randomized experiment that receives an inactive treatment but is otherwise managed exactly as the other groups
Convenience Sampling: a nonrandom method of selecting a sample; this method selects individuals that are easily accessible and may result in biased data.
Cumulative Relative Frequency: The term applies to an ordered set of observations from smallest to largest. The cumulative relative frequency is the sum of the relative frequencies for all values that are less than or equal to the given value.
Data: a set of observations (a set of possible outcomes); most data can be put into two groups: qualitative (an attribute whose value is indicated by a label) or quantitative (an attribute whose value is indicated by a number). Quantitative data can be separated into two subgroups: discrete and continuous. Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage)
Double-blind experiment: an experiment in which both the subjects of an experiment and the researchers who work with the subjects are blinded
Experimental Unit: any individual or object to be measured
Explanatory Variable: the independent variable in an experiment; the value controlled by researchers
Frequency: the number of times a value of the data occurs
Informed Consent: Any human subject in a research study must be cognizant of any risks or costs associated with the study. The subject has the right to know the nature of the treatments included in the study, their potential risks, and their potential benefits. Consent must be given freely by an informed, fit participant.
Institutional Review Board: a committee tasked with oversight of research programs that involve human subjects
Lurking Variable: a variable that has an effect on a study even though it is neither an explanatory variable nor a response variable
Nonsampling Error: an issue that affects the reliability of sampling data other than natural variation; it includes a variety of human errors including poor study design, biased sampling methods, inaccurate information provided by study participants, data entry errors, and poor analysis.
Numerical Variable: variables that take on values that are indicated by numbers
Parameter: a number that is used to represent a population characteristic and that generally cannot be determined easily
Placebo: an inactive treatment that cannot directly affect the response variable, used to counter the power of suggestion
Population: all individuals, objects, or measurements whose properties are being studied
Probability: a number between zero and one, inclusive, that gives the likelihood that a specific event will occur
Proportion: the number of successes divided by the total number in the sample
Qualitative Data: a set of observations (a set of possible outcomes); qualitative data has an attribute whose value is indicated by a label
Quantitative Data: a set of observations (a set of possible outcomes); quantitative (an attribute whose value is indicated by a number) data can be separated into two subgroups: discrete and continuous. Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage).
Random Assignment: the act of organizing experimental units into treatment groups using random methods
Random Sampling: a method of selecting a sample that gives every member of the population an equal chance of being selected.
Relative Frequency: the ratio of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes
Representative Sample: a subset of the population that has the same characteristics as the population
Response Variable: the dependent variable in an experiment; the value that is measured for change at the end of an experiment
Sample: a subset of the population studied
Sampling Bias: not all members of the population are equally likely to be selected
Sampling Error: the natural variation that results from selecting a sample to represent a larger population; this variation decreases as the sample size increases, so selecting larger samples reduces sampling error.
Sampling with Replacement: Once a member of the population is selected for inclusion in a sample, that member is returned to the population for the selection of the next individual.
Sampling without Replacement: A member of the population may be chosen for inclusion in a sample only once. If chosen, the member is not returned to the population before the next selection.
Simple Random Sampling: a straightforward method for selecting a random sample; give each member of the population a number. Use a random number generator to select a set of labels. These randomly selected labels identify the members of your sample.
Statistic: a numerical characteristic of the sample; a statistic estimates the corresponding population parameter.
Stratified Sampling: a method for selecting a random sample used to ensure that subgroups of the population are represented adequately; divide the population into groups (strata). Use simple random sampling to identify a proportionate number of individuals from each stratum.
Systematic Sampling: a method for selecting a random sample; list the members of the population. Use simple random sampling to select a starting point in the population. Let k = (number of individuals in the population)/(number of individuals needed in the sample). Choose every kth individual in the list starting with the one that was randomly selected. If necessary, return to the beginning of the population list to complete your sample.
Treatments: different values or components of the explanatory variable applied in an experiment
Variable: a characteristic of interest for each person or object in a population

Chapter 2

Descriptive statistics

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Summary

Before any inference, you picture the sample and summarize it. Stemplots, histograms, and box plots show shape. The five-number summary is the minimum, first quartile, median, third quartile, and maximum. The interquartile range is Q3 − Q1, and the 1.5 IQR fences flag outliers. The sample standard deviation divides the squared deviations by n − 1. The median resists outliers; the mean does not. A long right tail usually pulls the mean above the median.

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Box plot: a graph that gives a quick picture of the middle 50% of the data
First Quartile: the value that is the median of the of the lower half of the ordered data set
Frequency: the number of times a value of the data occurs
Frequency Polygon: looks like a line graph but uses intervals to display ranges of large amounts of data
Frequency Table: a data representation in which grouped data is displayed along with the corresponding frequencies
Histogram: a graphical representation in x-y form of the distribution of data in a data set; x represents the data and y represents the frequency, or relative frequency. The graph consists of contiguous rectangles.
Interquartile Range: or IQR, is the range of the middle 50 percent of the data values; the IQR is found by subtracting the first quartile from the third quartile.
Interval: also called a class interval; an interval represents a range of data and is used when displaying large data sets
Mean: a number that measures the central tendency of the data; a common name for mean is 'average.' The term 'mean' is a shortened form of 'arithmetic mean.' By definition, the mean for a sample (denoted by x¯) is x ¯ = Sum of all values in the sample Number of values in the sample , and the mean for a population (denoted by μ) is μ= Sum of all values in the population Number of values in the population .
Median: a number that separates ordered data into halves; half the values are the same number or smaller than the median and half the values are the same number or larger than the median. The median may or may not be part of the data.
Midpoint: the mean of an interval in a frequency table
Mode: the value that appears most frequently in a set of data
Outlier: an observation that does not fit the rest of the data
Paired Data Set: two data sets that have a one to one relationship so that: both data sets are the same size, and each data point in one data set is matched with exactly one point from the other set.
Percentile: a number that divides ordered data into hundredths; percentiles may or may not be part of the data. The median of the data is the second quartile and the 50th percentile. The first and third quartiles are the 25th and the 75th percentiles, respectively.
Quartiles: the numbers that separate the data into quarters; quartiles may or may not be part of the data. The second quartile is the median of the data.
Relative Frequency: the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes
Skewed: used to describe data that is not symmetrical; when the right side of a graph looks “chopped off” compared the left side, we say it is “skewed to the left.” When the left side of the graph looks “chopped off” compared to the right side, we say the data is “skewed to the right.” Alternatively: when the lower values of the data are more spread out, we say the data are skewed to the left. When the greater values are more spread out, the data are skewed to the right.
Standard Deviation: a number that is equal to the square root of the variance and measures how far data values are from their mean; notation: s for sample standard deviation and σ for population standard deviation.
Variance: mean of the squared deviations from the mean, or the square of the standard deviation; for a set of data, a deviation can be represented as x – x ¯ where x is a value of the data and x ¯ is the sample mean. The sample variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and one.

Chapter 3

Probability topics

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Summary

An event is a collection of outcomes, and its probability sits between 0 and 1. The complement is 1 minus the probability. The general addition rule subtracts the overlap so it is not counted twice. Conditional probability reads “given,” and independence means the joint probability factors into a product. Contingency tables hold the counts those rules use.

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AND Event: An outcome is in the event A AND B if the outcome is in both A AND B at the same time.
Complement Event: The complement of event A consists of all outcomes that are NOT in A.
Conditional Probability: the likelihood that an event will occur given that another event has already occurred
Conditional Probability of A GIVEN B: P(A|B) is the probability that event A will occur given that the event B has already occurred.
Conditional Probability of One Event Given Another Event: P(A|B) is the probability that event A will occur given that the event B has already occurred.
contingency table: the method of displaying a frequency distribution as a table with rows and columns to show how two variables may be dependent (contingent) upon each other; the table provides an easy way to calculate conditional probabilities.
Dependent Events: If two events are NOT independent, then we say that they are dependent.
Equally Likely: Each outcome of an experiment has the same probability.
Event: a subset of the set of all outcomes of an experiment; the set of all outcomes of an experiment is called a sample space and is usually denoted by S. An event is an arbitrary subset in S. It can contain one outcome, two outcomes, no outcomes (empty subset), the entire sample space, and the like. Standard notations for events are capital letters such as A, B, C, and so on.
Experiment: a planned activity carried out under controlled conditions
Independent Events: The occurrence of one event has no effect on the probability of the occurrence of another event. Events A and B are independent if one of the following is true: P(A|B) = P(A) P(B|A) = P(B) P(A AND B) = P(A)P(B)
Mutually Exclusive: Two events are mutually exclusive if the probability that they both happen at the same time is zero. If events A and B are mutually exclusive, then P(A AND B) = 0.
Or Event: An outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B.
Outcome: a particular result of an experiment
Probability: a number between zero and one, inclusive, that gives the likelihood that a specific event will occur; the foundation of statistics is given by the following 3 axioms (by A.N. Kolmogorov, 1930’s): Let S denote the sample space and A and B are two events in S. Then: 0 ≤ P(A) ≤ 1 If A and B are any two mutually exclusive events, then P(A OR B) = P(A) + P(B). P(S) = 1
Sample Space: the set of all possible outcomes of an experiment
Tree Diagram: the useful visual representation of a sample space and events in the form of a “tree” with branches marked by possible outcomes together with associated probabilities (frequencies, relative frequencies)
Venn Diagram: the visual representation of a sample space and events in the form of circles or ovals showing their intersections

Chapter 4

Discrete random variables

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Summary

A discrete random variable has a list of separate values, and its mean is the sum of each value times its probability. A binomial setting has a fixed number of independent trials, two outcomes, and a constant success probability; its mean is np and its variance is npq. Geometric waits for the first success. Hypergeometric draws without replacement. Poisson counts events over an interval of time or space.

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Bernoulli Trials: an experiment with the following characteristics: There are only two possible outcomes called “success” and “failure” for each trial. The probability p of a success is the same for any trial (so the probability q = 1 − p of a failure is the same for any trial).
Binomial Experiment: a statistical experiment that satisfies the following three conditions: There are a fixed number of trials, n. There are only two possible outcomes, called "success" and, "failure," for each trial. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial. The n trials are independent and are repeated using identical conditions.
Binomial Probability Distribution: a discrete random variable (RV) that arises from Bernoulli trials; there are a fixed number, n, of independent trials. “Independent” means that the result of any trial (for example, trial one) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV X is defined as the number of successes in n trials. The notation is: X ~ B(n, p). The mean is μ = np and the standard deviation is σ = npq . The probability of exactly x successes in n trials is P(X = x) = ( n x ) pxqn − x.
Expected Value: expected arithmetic average when an experiment is repeated many times; also called the mean. Notations: μ. For a discrete random variable (RV) with probability distribution function P(x),the definition can also be written in the form μ = ∑ xP(x).
Geometric Distribution: a discrete random variable (RV) that arises from the Bernoulli trials; the trials are repeated until the first success. The geometric variable X is defined as the number of trials until the first success. Notation: X ~ G(p). The mean is μ = 1 p and the standard deviation is σ = 1 p ( 1 p −1 ) . The probability of exactly x failures before the first success is given by the formula: P(X = x) = p(1 – p)x – 1.
Geometric Experiment: a statistical experiment with the following properties: There are one or more Bernoulli trials with all failures except the last one, which is a success. In theory, the number of trials could go on forever. There must be at least one trial. The probability, p, of a success and the probability, q, of a failure do not change from trial to trial.
Hypergeometric Experiment: a statistical experiment with the following properties: You take samples from two groups. You are concerned with a group of interest, called the first group. You sample without replacement from the combined groups. Each pick is not independent, since sampling is without replacement. You are not dealing with Bernoulli Trials.
Hypergeometric Probability: a discrete random variable (RV) that is characterized by: A fixed number of trials. The probability of success is not the same from trial to trial. We sample from two groups of items when we are interested in only one group. X is defined as the number of successes out of the total number of items chosen. Notation: X ~ H(r, b, n), where r = the number of items in the group of interest, b = the number of items in the group not of interest, and n = the number of items chosen.
Mean: a number that measures the central tendency; a common name for mean is ‘average.’ The term ‘mean’ is a shortened form of ‘arithmetic mean.’ By definition, the mean for a sample (detonated by x ¯ ) is x ¯ = Sum of all values in the sampleNumber of values in the sample and the mean for a population (denoted by μ) is μ = Sum of all values in the population Number of values in the population .
Mean of a Probability Distribution: the long-term average of many trials of a statistical experiment
Poisson Probability Distribution: a discrete random variable (RV) that counts the number of times a certain event will occur in a specific interval; characteristics of the variable: The probability that the event occurs in a given interval is the same for all intervals. The events occur with a known mean and independently of the time since the last event. The distribution is defined by the mean μ of the event in the interval. Notation: X ~ P(μ). The mean is μ = np. The standard deviation is σ = μ . The probability of having exactly x successes in r trials is P(X = x ) = ( e −μ ) μ x x! . The Poisson distribution is often used to approximate the binomial distribution, when n is “large” and p is “small” (a general rule is that n should be greater than or equal to 20 and p should be less than or equal to 0.05).
Probability Distribution Function (PDF): a mathematical description of a discrete random variable (RV), given either in the form of an equation (formula) or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome.
Random Variable (RV): a characteristic of interest in a population being studied; common notation for variables are upper case Latin letters X, Y, Z,...; common notation for a specific value from the domain (set of all possible values of a variable) are lower case Latin letters x, y, and z. For example, if X is the number of children in a family, then x represents a specific integer 0, 1, 2, 3,.... Variables in statistics differ from variables in intermediate algebra in the two following ways. The domain of the random variable (RV) is not necessarily a numerical set; the domain may be expressed in words; for example, if X = hair color then the domain is {black, blond, gray, green, orange}. We can tell what specific value x the random variable X takes only after performing the experiment.
Standard Deviation of a Probability Distribution: a number that measures how far the outcomes of a statistical experiment are from the mean of the distribution σ=∑[x – μ2 ∙ Ρx]
The Law of Large Numbers: As the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency probability approaches zero.

Chapter 5

Continuous random variables

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Summary

A continuous variable is measured, so probability is the area under a density curve, and the whole area is 1. A single exact number has no width and therefore probability 0. The uniform density is flat between two endpoints, so probability is just the length of the interval divided by the total length. The exponential distribution is the other model in this chapter, used for waiting times.

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Conditional Probability: the likelihood that an event will occur given that another event has already occurred.
decay parameter: The decay parameter describes the rate at which probabilities decay to zero for increasing values of x. It is the value m in the probability density function f(x) = me(-mx) of an exponential random variable. It is also equal to m = 1 μ , where μ is the mean of the random variable.
Exponential Distribution: a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital; the notation is X ~ Exp(m). The mean is μ = 1 m and the standard deviation is σ = 1 m . The probability density function is f(x) = me−mx, x ≥ 0 and the cumulative distribution function is P(X ≤ x) = 1 − e−mx.
memoryless property: For an exponential random variable X, the memoryless property is the statement that knowledge of what has occurred in the past has no effect on future probabilities. This means that the probability that X exceeds x + k, given that it has exceeded x, is the same as the probability that X would exceed k if we had no knowledge about it. In symbols we say that P(X > x + k|X > x) = P(X > k).
Poisson distribution: If there is a known average of λ events occurring per unit time, and these events are independent of each other, then the number of events X occurring in one unit of time has the Poisson distribution. The probability of k events occurring in one unit time is equal to P(X=k)= λ k e −λ k! .
Uniform Distribution: a continuous random variable (RV) that has equally likely outcomes over the domain, a < x < b. Notation: X ~ U(a,b). The mean is μ = a+b 2 and the standard deviation is σ= ( b−a ) 2 12 . The probability density function is f(x) = 1 b−a for a < x < b or a ≤ x ≤ b. The cumulative distribution is P(X ≤ x) = x−a b−a .

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Continuous random variable: A variable whose values are measured and can fall anywhere in an interval, not only on a list of separate numbers.
Probability density function: A curve used for a continuous variable. Probability is the area under the curve, and the total area is 1.
P(X = c) = 0: For a continuous variable, one exact value has no width, so it has no area and probability 0.

Chapter 6

The normal distribution

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Summary

The normal curve is the symmetric bell centered at the mean. A z-score counts standard deviations from that mean, z = (x − μ) / σ, and the standard normal curve has mean 0 and standard deviation 1. The empirical rule puts about 68% of the data within one standard deviation, 95% within two, and 99.7% within three. Negative z-scores sit below the mean.

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Normal Distribution: a continuous random variable (RV) with pdf f(x) = 1 σ 2π e –(x – μ) 2 σ 2 2 , where μ is the mean of the distribution and σ is the standard deviation; notation: X ~ N(μ, σ). If μ = 0 and σ = 1, the RV is called the standard normal distribution.
Standard Normal Distribution: a continuous random variable (RV) X ~ N(0, 1); when X follows the standard normal distribution, it is often noted as Z ~ N(0, 1).
z-score: the linear transformation of the form z = x – μ σ ; if this transformation is applied to any normal distribution X ~ N(μ, σ) the result is the standard normal distribution Z ~ N(0,1). If this transformation is applied to any specific value x of the RV with mean μ and standard deviation σ, the result is called the z-score of x. The z-score allows us to compare data that are normally distributed but scaled differently.

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Empirical rule: For a normal distribution, about 68% of the values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

Chapter 7

The central limit theorem

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Summary

A sample mean is a random variable of its own. Its sampling distribution is centered at the population mean μ, with standard error σ / √n, so larger samples make the mean less variable. If the population is normal, the sample mean is normal for every sample size. If it is not, the central limit theorem says the sample mean is still approximately normal once n is large — the text often uses 30 as a guide.

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Average: a number that describes the central tendency of the data; there are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean.
Central Limit Theorem: Given a random variable (RV) with known mean μ and known standard deviation, σ, we are sampling with size n, and we are interested in two new RVs: the sample mean, X ¯ , and the sample sum, ΣΧ. If the size (n) of the sample is sufficiently large, then X ¯ ~ N(μ, σ n ) and ΣΧ ~ N(nμ, ( n )(σ)). If the size (n) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distributions regardless of the shape of the population. The mean of the sample means will equal the population mean, and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means, σ n , is called the standard error of the mean.
Exponential Distribution: a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital, notation: X ~ Exp(m). The mean is μ = 1 m and the standard deviation is σ = 1 m . The probability density function is f(x) = me–mx, x ≥ 0 and the cumulative distribution function is P(X ≤ x) = 1 – e–mx.
Mean: a number that measures the central tendency; a common name for mean is "average." The term "mean" is a shortened form of "arithmetic mean." By definition, the mean for a sample (denoted by x ¯ ) is x ¯ = Sum of all values in the sample Number of values in the sample , and the mean for a population (denoted by μ) is μ = Sum of all values in the population Number of values in the population .
Normal Distribution: a continuous random variable (RV) with pdf f(x) = 1 σ 2π e – (x – μ) 2 2 σ 2 , where μ is the mean of the distribution and σ is the standard deviation; notation: Χ ~ N(μ, σ). If μ = 0 and σ = 1, the RV is called a standard normal distribution.
Sampling Distribution: Given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution.
Standard Error of the Mean: the standard deviation of the distribution of the sample means, or σ n .
Uniform Distribution: a continuous random variable (RV) that has equally likely outcomes over the domain, a < x < b; often referred as the Rectangular Distribution because the graph of the pdf has the form of a rectangle. Notation: X ~ U(a, b). The mean is μ = a + b 2 and the standard deviation is σ = (b–a) 2 12 . The probability density function is f(x) = 1 b–a for a < x < b or a ≤ x ≤ b. The cumulative distribution is P(X ≤ x) = x–a b–a .

Chapter 8

Confidence intervals

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Summary

A confidence interval is a point estimate plus or minus a margin of error. The percent is about the method: in the long run, 95% of such intervals capture the parameter. It is not a probability that this finished interval contains the mean. Unknown σ uses a t critical value with n − 1 degrees of freedom. Raising the confidence level widens the interval; raising n narrows it. A proportion interval needs enough successes and failures, at least 5 of each in this text.

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Binomial Distribution: a discrete random variable (RV) which arises from Bernoulli trials; there are a fixed number, n, of independent trials. “Independent” means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV X is defined as the number of successes in n trials. The notation is: X~B(n,p). The mean is μ = np and the standard deviation is σ = npq . The probability of exactly x successes in n trials is P ( X = x ) = n x p x q n − x .
Confidence Interval (CI): an interval estimate for an unknown population parameter. This depends on: the desired confidence level, information that is known about the distribution (for example, known standard deviation), the sample and its size.
Confidence Level (CL): the percent expression for the probability that the confidence interval contains the true population parameter; for example, if the CL = 90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter.
Degrees of Freedom (df): the number of objects in a sample that are free to vary
Error Bound for a Population Mean (EBM): the margin of error; depends on the confidence level, sample size, and known or estimated population standard deviation.
Inferential Statistics: also called statistical inference or inductive statistics; this facet of statistics deals with estimating a population parameter based on a sample statistic. For example, if four out of the 100 calculators sampled are defective we might infer that four percent of the production is defective.
Normal Distribution: a continuous random variable (RV) with pdf f(x)= 1 σ 2π e – (x–μ) 2 /2 σ 2 , where μ is the mean of the distribution and σ is the standard deviation, notation: X ~ N(μ,σ). If μ = 0 and σ = 1, the RV is called the standard normal distribution.
Parameter: a numerical characteristic of a population
Point Estimate: a single number computed from a sample and used to estimate a population parameter
Standard Deviation: a number that is equal to the square root of the variance and measures how far data values are from their mean; notation: s for sample standard deviation and σ for population standard deviation
Student's t-Distribution: investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student; the major characteristics of the random variable (RV) are: It is continuous and assumes any real values. The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution. It approaches the standard normal distribution as n get larger. There is a "family" of t–distributions: each representative of the family is completely defined by the number of degrees of freedom, which is one less than the number of data.

Chapter 9

Hypothesis testing with one sample

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Summary

A test sets a null hypothesis against an alternative, then asks whether the data are surprising if the null is true. That surprise is the p-value. You reject the null when the p-value is at most α. A large p-value means you fail to reject; it does not prove the null. Rejecting a true null is a Type I error, and its rate is α. Failing to reject a false null is a Type II error. The form of the alternative decides a left-tailed, right-tailed, or two-tailed test.

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Binomial Distribution: a discrete random variable (RV) that arises from Bernoulli trials. There are a fixed number, n, of independent trials. “Independent” means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV Χ is defined as the number of successes in n trials. The notation is: X ~ B(n, p) μ = np and the standard deviation is σ= npq . The probability of exactly x successes in n trials is P(X=x)=( n x ) p x q n−x .
Central Limit Theorem: Given a random variable (RV) with known mean μ and known standard deviation σ. We are sampling with size n and we are interested in two new RVs - the sample mean, X ¯ , and the sample sum, ΣX. If the size n of the sample is sufficiently large, then X ¯ ~N( μ, σ n ) and ΣX~N(nμ, n σ) . If the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means, σ n , is called the standard error of the mean.
Confidence Interval (CI): an interval estimate for an unknown population parameter. This depends on: The desired confidence level. Information that is known about the distribution (for example, known standard deviation). The sample and its size.
Hypothesis: a statement about the value of a population parameter, in case of two hypotheses, the statement assumed to be true is called the null hypothesis (notation H0) and the contradictory statement is called the alternative hypothesis (notation Ha).
Hypothesis Testing: Based on sample evidence, a procedure for determining whether the hypothesis stated is a reasonable statement and should not be rejected, or is unreasonable and should be rejected.
Level of Significance of the Test: probability of a Type I error (reject the null hypothesis when it is true). Notation: α. In hypothesis testing, the Level of Significance is called the preconceived α or the preset α.
Normal Distribution: a continuous random variable (RV) with pdf f(x)= 1 σ 2π e − (x−μ) 2 2 σ 2 , where μ is the mean of the distribution, and σ is the standard deviation, notation: X ~ N(μ, σ). If μ = 0 and σ = 1, the RV is called the standard normal distribution.
p-value: the probability that an event will happen purely by chance assuming the null hypothesis is true. The smaller the p-value, the stronger the evidence is against the null hypothesis.
Standard Deviation: a number that is equal to the square root of the variance and measures how far data values are from their mean; notation: s for sample standard deviation and σ for population standard deviation.
Student's t-Distribution: investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student. The major characteristics of the random variable (RV) are: It is continuous and assumes any real values. The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution. It approaches the standard normal distribution as n gets larger. There is a "family" of t distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data items.
Type I Error: The decision is to reject the null hypothesis when, in fact, the null hypothesis is true.
Type II Error: The decision is not to reject the null hypothesis when, in fact, the null hypothesis is false.

Chapter 10

Hypothesis testing with two samples

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Summary

Two groups are independent when no observation in one is paired with an observation in the other. Before-and-after measurements on the same people are matched pairs, and the test is a one-sample test on the differences, often with null mean difference 0. Separate groups are compared with a two-mean or two-proportion procedure. A two-proportion test compares the success rates of two independent groups.

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Degrees of Freedom (df): the number of objects in a sample that are free to vary.
Pooled Proportion: estimate of the common value of p1 and p2.
Standard Deviation: A number that is equal to the square root of the variance and measures how far data values are from their mean; notation: s for sample standard deviation and σ for population standard deviation.
Variable (Random Variable): a characteristic of interest in a population being studied. Common notation for variables are upper-case Latin letters X, Y, Z,... Common notation for a specific value from the domain (set of all possible values of a variable) are lower-case Latin letters x, y, z,.... For example, if X is the number of children in a family, then x represents a specific integer 0, 1, 2, 3, .... Variables in statistics differ from variables in intermediate algebra in the two following ways. The domain of the random variable (RV) is not necessarily a numerical set; the domain may be expressed in words; for example, if X = hair color, then the domain is {black, blond, gray, green, orange}. We can tell what specific value x of the random variable X takes only after performing the experiment.

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Independent samples: Groups in which no observation in one sample is paired with an observation in the other.
Matched pairs: Two linked measurements, often before and after on the same person. The test uses the differences.
Mean difference: The average of the paired differences. It estimates the population mean difference μd.

Chapter 11

The chi-square distribution

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Summary

Chi-square tests compare counts. Goodness of fit asks whether category counts match a claimed distribution. Independence and homogeneity use a contingency table. The expected count in a cell is (row total × column total) / n, and each expected count should be at least 5. Degrees of freedom are k − 1 for a fit test and (r − 1)(c − 1) for a table. The statistic adds squared gaps between observed and expected counts, so the test is right-tailed.

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Contingency Table: a table that displays sample values for two different factors that may be dependent or contingent on one another; it facilitates determining conditional probabilities.

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Goodness-of-fit test: A chi-square test of whether observed category counts match one claimed distribution.
Test of independence: A chi-square test of whether two categorical variables in a contingency table are associated.
Expected count: The count a cell would have if the null were true. For a table, (row total × column total) / n.
Degrees of freedom: k − 1 for goodness of fit with k categories. (r − 1)(c − 1) for a table with r rows and c columns.
Chi-square statistic: A sum of (observed − expected)² / expected. Large values are the evidence, so the test is right-tailed.

Chapter 12

Linear regression and correlation

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Summary

The correlation r is a unitless number from −1 to 1. It measures linear association and does not by itself show that one variable causes the other. The least-squares line predicts y from x; the residual is the observed value minus that prediction. r² is the fraction of the variation in y accounted for by the line. Using the line far outside the observed x-values is extrapolation, and it can fail.

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Outlier: an observation that does not fit the rest of the data

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Correlation coefficient (r): A number from −1 to 1 that measures the direction and strength of a linear association. It does not prove cause.
Coefficient of determination (r²): The fraction of the variation in y accounted for by the linear relationship with x.
Least-squares line: The line that minimizes the sum of squared residuals.
Residual: Observed y minus the y predicted by the line.
Slope: The predicted change in y when x increases by one unit.
Extrapolation: Using the fitted line far outside the x-values that were observed.

Chapter 13

F distribution and one-way ANOVA

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Summary

One-way ANOVA compares the means of three or more groups in a single test. The null hypothesis says every group mean is equal; the alternative says at least one differs. The F statistic is the variance between groups divided by the variance within groups, and only a large F counts against the null, so the test is right-tailed. The usual assumptions are independent samples from normal populations with a common variance.

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Analysis of Variance: also referred to as ANOVA, is a method of testing whether or not the means of three or more populations are equal. The method is applicable if: all populations of interest are normally distributed. the populations have equal standard deviations. samples (not necessarily of the same size) are randomly and independently selected from each population. The test statistic for analysis of variance is the F-ratio.
One-Way ANOVA: a method of testing whether or not the means of three or more populations are equal; the method is applicable if: all populations of interest are normally distributed. the populations have equal standard deviations. samples (not necessarily of the same size) are randomly and independently selected from each population. there is one independent variable and one dependent variable. The test statistic for analysis of variance is the F-ratio.
Variance: mean of the squared deviations from the mean; the square of the standard deviation. For a set of data, a deviation can be represented as x – x ¯ where x is a value of the data and x ¯ is the sample mean. The sample variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and one.

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F statistic: Between-group variance divided by within-group variance.
Between-groups degrees of freedom: k − 1, one less than the number of groups.
Within-groups degrees of freedom: N − k, the total number of observations minus the number of groups.